Meta-analysis

9. Technical notes A. Meta-analysis According to Glass (28), a primary analysis is the original analysis of data in a research study. Secon- dary analysis is the re-analysis of data for the pur- pose of answering the original research questions with better statistical techniques, or answering new questions with old data. Meta-analysis refers to the analysis of analyses or "the statistical analysis of a large collection of analysis results from individual studies for the purpose of integrating the findings. It connotes a rigorous altemative to the casual, nar- rative discussions of research studies which typify our attempts to make sense of a large volume of re- search literature". In the context of the present project, original data sets were obtained and all analyses were conducted on these. Statistics (i.e., ORs) from individual studies were then combined formally by means of the Confidence Profile (CP) method. Meta-analysis by the Confidence Profile method Basically, as applied here, the Confidence Profile method (7) takes information on the parameter of interest from the available studies and, by means of a Bayesian model, derives a joint posterior probability distribution for this parameter. The process is illus- trated in the following. The observed result from the analysis of a single study is denoted as 0 (the parameter of interest, e.g., the odds ratio). The evidence supporting this parame- ter is supplied by the data. This may be the propor- tion of cases y - low-birth-weight infants, for exam- ple - in the study. The likelihood function for the evidence, given the parameter, is: L(y 0). If our belief in the parameter is expressed as a prior distribution for 0: 7(O), then we can calculate the posterior distribution as the product of the like- lihood function and the prior distribution: T(1|y) = k L(yI|);ir() If we have no strong belief in the value of the prior ditribution, then a non-informative prior can be used. In the CP method this is based on a beta distri- bution with parameters x = 1/2 and f = 1/2. The posterior distribution obtained from the evidence derived from our first study becomes the prior distri- bution for evaluating the second study: ,r( IY1 Y2) = k L(y21 9)Z(9 I Y1) And this gives the new posterior distribution based on both studies. The process can be extended in this manner to incorporate evidence from all available studies. The result is a posterior probability distribution which may be graphed and studied. B. Regression models for inter- study heterogeneity A number of references have been made in Chapters 2 and 4 on the need to test the odds ratios for homo- geneity before meta-analysis. This test is of the null hypothesis that all study ORs are estimating the same underlying value, versus the alternative that at least one of the ORs differs from the remainder (the 'Q' test for homogeneity, see Hedges & Olkin (10)). Should the test be significant, a weighted regression analysis may supply an explanation of the inter-study variation in the ORs. The regression model will include some of the study characteristics described in the study quality table at the end of Chapter 2, includ- ing study design (prospective versus retrospective), prevalence level of the outcome, country groupings, regional groups (Africa, southern Asia, south-east Asia, Latin America, USA and Europe), location of the study (urban, rural, mixed), decade of the study, WHO Collaborative study versus other, as well as maternal characteristics such as mean maternal age, mean gestational age at delivery, etc. As was noted in Chapter 4, many of the tests were confirmed as statistically significant, and these cases were exam- ined by regression analysis. Some of the results will be given in brief to provide an indication of the fac- tors having some explanatory power. Analysis for IUGR Predictor: attained weight at month 5. For this pre- dictor and outcome, the regression analysis accoun- ted for around 34% of the observed variation in the study ORs. The one variable found to be significant was the WHO study versus other study factor. This results from a significant difference in the means of the odds ratios for both categories. The reason is unlikely to be a different biological relationship be- tween predictor and outcome in the geographical areas covered by either study. Some bias to external validity is more reasonable. None of the remaining factors was statistically significant 48 WHO Bulletin OMS: Supplement Vol. 73 1995